Phase-Mapper features novel solver AgileFD as a
key component of the platform. Motivated by convolutive NMF, AgileFD includes a set of lightweight
updating rules, and therefore a very fast gradient
descent process. AgileFD is flexible, allowing for the
incorporation of additional contraints, as well as
human feedback through refinement. AgileFD can
also run autonomously, producing physically meaningful solutions.

Phase-Mapper also provides tools for data exploration, visualization, and configuration that allow
human experts as well as laypeople to analyze and
improve solutions.

Phase-Mapper’s solutions, obtained by the interaction between solvers and human users or
autonomously, can also shed light on the development of new physical experiments. For example, the
results can be incorporated into an active learning
system, specifying regions of composition space to
sample at higher resolution.

AgileFD: A NovelPhase-Mapping SolverThe Phase-Mapper platform features the AgileFDsolver for the phase-mapping problem. AgileFD usesiterative updates of candidate solutions that are sig-nificantly faster than previously proposed methods.Human experts can interact with the algorithm inreal time, and this speed is due to an efficient prob-lem representation. Let the XRD patterns for all sam-ples be represented by a matrix A, where each col-umn corresponds to one sample point and each rowcorresponds to Aj(q) for a particular value of q. Underthe assumptions of no noise and no shifting, mean-ing that λij = 1 for all i and j, describing A as a linearcombination of a few basis patterns Wi(q) is equiva-lent to factoring A as a product of two matrices Wand H:

( 2)

Here, R denotes the approximate reconstruction of A.In this formulation, the columns of W form a set ofbasis patterns Wi(q), and the columns of H corre-spond to the values hij in equation 1. We enforcenonnegativity for W and H, which is required for thesolutions to be physically meaningful. Previousapproaches to solve the phase-mapping problembased on NMF have been unsuccessful in handlingpeak shifting, where λij ≠ 1. The first contribution ofAgileFD is to circumvent the shifting problem by alog space resampling. Under the variable transforma-tion q into log q, our signal becomes Wi(log q). Moreimportantly, the shifted phase Wi(log λq) becomesWi(log λ + log q), which transforms the multiplicativeshift in the q domain into a constant additive offset.This allows the problem to be formulated in terms ofconvolutive nonnegative matrix factorization. Afterthis variable substitution, we discretize the values ofallowed λ and interpolate the signals at the corre-sponding geometric series of q values. The problemcan then be written:( 3)

With the columns of W representing the basis pat-

A ; W ;mm= 1

M

; ;Hm=RA;W;H =RArticles

Figure 3. An Illustration of the Phase-Mapping Problem.

Given a material system with XRD data read at discrete points, find a set of
basis phases, such that every point’s XRD data can be made by a linear combination of the basis phases. Here, the left image is the original data, the
right image is the found basis phases, and the middle image represents how
much of a particular phase (the “phase concentration”) is present at each
data point along with the composition-dependent shifting.